Name: transfer function gain and origin effects
Uploaded: 2024-07-08T13:20:02.000Z
Duration: 1 min 19 s
Channel: JoVE

Si consideri la funzione di trasferimento di un sistema espressa in forma standard.
La funzione di trasferimento ha un guadagno costante, uno zero e un polo all'origine, uno zero semplice, un polo semplice, un polo quadratico e uno zero quadratico.
I diagrammi di Bode per i singoli fattori vengono tracciati separatamente e combinati graficamente.
Per il termine di guadagno costante positivo, il guadagno di grandezza è costante e l'angolo di fase è zero e indipendente dalla frequenza.
Per il termine di guadagno costante negativo, il guadagno di magnitudine rimane lo stesso e l'angolo di fase è di ±180°.
Per una funzione di trasferimento con uno zero all'origine, il grafico della magnitudine rappresenta una retta con una pendenza positiva di 20 dB/decennio. Il grafico di fase mostra un angolo di fase costante di 90°.
Per un polo all'origine, la pendenza del grafico di magnitudine è di -20 dB/decade e lo sfasamento è di -90°.
Se all'origine è presente più di un polo o zero, la pendenza del grafico di magnitudine e lo sfasamento vengono moltiplicati per il numero di poli o zeri presenti all'origine.

risposta in frequenza

Una funzione di trasferimento presentata nella sua forma standard integra il guadagno costante degli elementi, gli zeri e i poli all'origine, zeri e poli semplici e poli e zeri quadratici. La funzione di trasferimento può essere scritta come H(ω):


<img alt="Equation 1" src="/files/ftp_upload/15141/15141_Equation_1.svg" style="height: 40px;">

La funzione di trasferimento, H(ω), spesso espressa nella forma standard, viene derivata normalizzando i coefficienti polinomiali della funzione di trasferimento. I poli (jω) e gli zeri (jω) sono frequenze critiche in cui l'entità e la fase dell'output del sistema subiscono cambiamenti significativi.


Guadagno, K:

La funzione di trasferimento ha un termine costante K, con un modulo di 20 log_10K e un angolo di fase di 0° per un valore positivo di K. Sia il modulo che la fase sono costanti con la frequenza. Se K è negativo, la grandezza rimane invariata, ma la fase è ±180°. Quando il guadagno K = 1, la grandezza diventerà zero insieme all'angolo di fase. In termini di diagramma di Bode, la grandezza è espressa in decibel (dB) come 20log_10K e la sua fase rimane a 0° o 180° a seconda del segno di K.

Polo/Zero all'Origine:

Un polo o uno zero all'origine hanno un impatto determinante sulla trama. Uno zero (jω)^+1 nell'origine ha modulo 20 log_10 ω e la fase è 90°. La pendenza del grafico della magnitudo è 20 dB/dacade e la fase è costante con la frequenza.

Per un polo (jω)^-1 all'origine, l'intensità è -20 dB/decade e la fase è -90°. Generalmente, per (jω)^N, dove N è un numero intero, il grafico della magnitudo avrà una pendenza di -20NdB/decade, mentre la fase è 90N°.

La grandezza costante K, i poli/zeri alle origini hanno l'angolo di fase che non cambia con la frequenza. Nel caso degli zeri/poli, l'angolo di fase cambia quando cambia il numero di poli/zeri nell'origine.

Risposta in frequenza

Si consideri la funzione di trasferimento di un sistema espressa in forma standard.
La funzione di trasferimento ha un guadagno costante, uno zero e un polo all'origine, un ...

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Transfer function and Bode Plots-I

transfer function gain and origin effects

Transfer function gain and poles or zeros at the origin shape a system's Bode plot. The transfer function is often written in standard form after the polynomial coefficients are normalized. In that form, the constant gain, zeros, poles, and quadratic terms all appear clearly. The poles and zeros mark critical frequencies where the output magnitude and phase change noticeably.

The constant gain K creates a fixed magnitude and phase term. Its magnitude is 20 log 10 K in decibels, and its phase is 0° when K is positive. If K is negative, the magnitude stays the same, but the phase becomes ±180°. When K = 1, the magnitude is 0 dB and the phase angle is 0°.

A zero at the origin has a strong effect on the Bode plot. For a zero at the origin, the magnitude is 20 log 10 ω, and the phase is 90°. The magnitude rises with a slope of 20 dB/decade, and the phase stays constant as frequency changes. A pole at the origin acts in the opposite way. Its magnitude slope is -20 dB/decade, and its phase is -90°.

More generally, a factor of (jω)^N changes the plot by an integer amount. For a zero at the origin, the magnitude slope becomes 20N dB/decade and the phase is 90N°. For a pole at the origin, the magnitude slope becomes -20N dB/decade, and the phase is also 90N°. In both cases, the constant gain K and the poles or zeros at the origin keep a phase angle that does not change with frequency. The phase changes when the number of poles or zeros at the origin changes.

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Explore how constant gain, poles, and zeros at the origin affect transfer functions and Bode plots, including dB slope and phase changes.

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JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Transfer Function Gain and Origin Effects

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